Since \({\rm Gr}_{k,n}^{>0}\) , \({\rm Mat}_{n,k+m}^{>0}\)  are path connected (see [1]} for the positive Grassmannian) and \({\rm Gr}_{k,n}^{\ge 0}=\overline{{\rm Gr}_{k,n}}\) , there exists a path
\(\left(\breve{C},\breve{\mathcal {Z}}\right):\left[0,1\right]\rightarrow {\rm Gr}_{k,n}^{\ge 0}\times {\rm Mat}_{n,k+m}^{>0}\) 
